Average Error: 29.0 → 0.4
Time: 2.7s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -6.7147976039290032 \cdot 10^{-6}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \le -6.7147976039290032 \cdot 10^{-6}:\\
\;\;\;\;e^{a \cdot x} - 1\\

\mathbf{else}:\\
\;\;\;\;a \cdot x + a \cdot \left(x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\\

\end{array}
double code(double a, double x) {
	return ((double) (((double) exp(((double) (a * x)))) - 1.0));
}
↓
double code(double a, double x) {
	double VAR;
	if ((((double) (a * x)) <= -6.714797603929003e-06)) {
		VAR = ((double) (((double) exp(((double) (a * x)))) - 1.0));
	} else {
		VAR = ((double) (((double) (a * x)) + ((double) (a * ((double) (x * ((double) (a * ((double) (x * 0.5))))))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.1
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* a x) < -6.7147976039290032e-6

    1. Initial program 0.1

      \[e^{a \cdot x} - 1\]

    if -6.7147976039290032e-6 < (* a x)

    1. Initial program 44.2

      \[e^{a \cdot x} - 1\]

    2. Taylor expanded around 0 14.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified7.9

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(a \cdot \left(a \cdot \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {a}^{3}\right)\right)\right)}\]

    4. Taylor expanded around 0 8.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + a \cdot x}\]

    5. Simplified4.3

      \[\leadsto \color{blue}{a \cdot x + a \cdot \left(\left(a \cdot \frac{1}{2}\right) \cdot \left(x \cdot x\right)\right)}\]
    6. Using strategy rm

    7. Applied associate-*r*0.6

      \[\leadsto a \cdot x + a \cdot \color{blue}{\left(\left(\left(a \cdot \frac{1}{2}\right) \cdot x\right) \cdot x\right)}\]

    8. Simplified0.6

      \[\leadsto a \cdot x + a \cdot \left(\color{blue}{\left(a \cdot \left(\frac{1}{2} \cdot x\right)\right)} \cdot x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -6.7147976039290032 \cdot 10^{-6}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(x \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))